Problem 32 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ \frac{5 y}{2} \geq 15 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and find the range of values for which the inequality holds true: $$ \frac{5 y}{2} \geq 15 $$ Answer: The solution for the inequality is $$ y \geq 6 $$, meaning the range of values for which the inequality holds true are all values greater than or equal to 6.

Step 1: Write down the inequality

We start with the inequality given: $$ \frac{5 y}{2} \geq 15 $$

Step 2: Remove the fraction

To remove the fraction from the inequality, we will multiply both sides by the denominator (2). This will not change the inequality symbol since 2 is a positive value: $$ 2 \cdot \frac{5 y}{2} \geq 15 \cdot 2 $$ Which simplifies to: $$ 5y \geq 30 $$

Step 3: Solve for "y"

Now we will isolate the variable "y" by dividing both sides by the coefficient 5: $$ \frac{5y}{5} \geq \frac{30}{5} $$ Which simplifies to: $$ y \geq 6 $$

Step 4: Write down the solution

The solution for the inequality is: $$ y \geq 6 $$

Key Concepts

Algebraic Inequalities

Algebraic inequalities are similar to equations but instead of an equals sign, they use inequality symbols to show the relationship between two expressions. Inequalities are fundamental in expressing ranges of possible values rather than a single value. When solving inequalities such as \( \frac{5y}{2} \geq 15 \), our main goal is to find all possible values of the variable that make the inequality true.

Improving your understanding of inequalities begins with recognizing that they represent a set of answers, not just one. For example, if given \( y \geq 6 \), this means that any number greater than or equal to 6 is a solution, including 6 itself. This could be numbers like 7, 10, or 1000. By picturing a number line, we see that all points to the right of 6, including 6, satisfy the inequality.

Inequality Symbols

Inequality symbols are the shorthand used to compare two quantities. There are four common inequality symbols:\(<, >, \leq, \geq\). The ‘less than’ (\(<\)) and ‘greater than’ (\(>\)) symbols are used when one value is strictly smaller or larger than another. On the other hand, ‘less than or equal to’ (\(\leq\)) and ‘greater than or equal to’ (\(\geq\)) incorporate equality, meaning the values could be unequal or exactly equal.

For instance, in the problem \(\frac{5y}{2} \geq 15\), the symbol \(\geq\) tells us that the value of \(5y\) divided by 2 is either greater than or exactly 15. These symbols are crucial for understanding the range of solutions. Remembering which way they 'point' can help you remember their meaning: the small end points to the smaller value and the wide open end to the larger value.

Isolating Variables

Isolating variables is a method where we aim to get the variable of interest by itself on one side of the inequality (or equation) to find the solution set. In solving \( \frac{5y}{2} \geq 15 \) we want to 'isolate' the variable \(y\), getting it on one side of the inequality by itself.

To isolate variables properly, we perform the same operation on both sides of an inequality without changing its sense. In the given problem, multiplying both sides by 2 eliminates the denominator, and then dividing by 5 isolates \(y\). It's crucial not to multiply or divide by negative numbers without flipping the inequality symbol, which is an essential rule in inequalities that differs from standard equations.